Systematic Constructions of Bent-Negabent Functions, 2-Rotation Symmetric Bent-Negabent Functions and Their Duals

Bent-negabent functions have many important properties for their application in cryptography since they have the flat absolute spectrum under the both Walsh-Hadamard transform and nega-Hadamard transform. In this paper, we present four new systematic constructions of bent-negabent functions on $4k, 8k, 4k+2$ and $8k+2$ variables, respectively, by modifying the truth tables of two classes of quadratic bent-negabent functions with simple form. The algebraic normal forms and duals of these constructed functions are also determined. We further identify necessary and sufficient conditions for those bent-negabent functions which have the maximum algebraic degree. At last, by modifying the truth tables of a class of quadratic 2-rotation symmetric bent-negabent functions, we present a construction of 2-rotation symmetric bent-negabent functions with any possible algebraic degrees. Considering that there are probably no bent-negabent functions in the rotation symmetric class, it is the first significant attempt to construct bent-negabent functions in the generalized rotation symmetric class

Secondary constructions of vectorial pp-ary weakly regular bent functions

In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of vectorial/Boolean bent functions via the so-called $(P_U)$ property was introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the construction of $p$-ary weakly regular bent functions. The objective of this paper is to further generalize these constructions, following the ideas in \cite{Bapic, Zheng}, for secondary constructions of vectorial $p$-ary weakly regular bent and plateaued functions. We also present some infinite families of such functions via the $p$-ary Maiorana-McFarland class. Additionally, we give another characterization of the $(P_U)$ property for the $p$-ary case via second-order derivatives, as it was done for the Boolean case in \cite{Zheng}

A new method for secondary constructions of vectorial bent functions

In 2017, Tang et al. have introduced a generic construction for bent functions of the form $f(x)=g(x)+h(x)$, where $g$ is a bent function satisfying some conditions and $h$ is a Boolean function. Recently, Zheng et al. generalized this result to construct large classes of bent vectorial Boolean function from known ones in the form $F(x)=G(x)+h(X)$, where $G$ is a bent vectorial and $h$ a Boolean function. In this paper we further generalize this construction to obtain vectorial bent functions of the form $F(x)=G(x)+\mathbf{H}(X)$, where $\mathbf{H}$ is also a vectorial Boolean function. This allows us to construct new infinite families of vectorial bent functions, EA-inequivalent to $G$, which was used in the construction. Most notably, specifying $\mathbf{H } (x)=\mathbf{h} (Tr_1^n(u_1x),\ldots,Tr_1^n(u_tx))$, the function $\mathbf{h} :\mathbb{F}_2^t \rightarrow \mathbb{F}_{2^t}$ can be chosen arbitrary which gives a relatively large class of different functions for a fixed function $G$. We also propose a method of constructing vectorial $(n,n)$-functions having maximal number of bent components

Composition construction of new bent functions from known dually isomorphic bent functions

Bent functions are optimal combinatorial objects and have been studied over the last four decades. Secondary construction plays a central role in constructing bent functions since it may generate bent functions outside the primary classes of bent functions. In this study, we improve a theoretical framework of the secondary construction of bent functions in terms of the composition of Boolean functions. Based on this framework, we propose several constructions of bent functions through the composition of a balanced Boolean function and dually isomorphic (DI) bent functions defined herein. In addition, we present a construction of self-dual bent functions

Constructing new superclasses of bent functions from known ones

Some recent research articles [23, 24] addressed an explicit specification of indicators that specify bent functions in the so-called $\mathcal{C}$ and $\mathcal{D}$ classes, derived from the Maiorana- McFarland ($\mathcal{M}$) class by C. Carlet in 1994 [5]. Many of these bent functions that belong to $\mathcal{C}$ or $\mathcal{D}$ are provably outside the completed $\mathcal{M}$ class. Nevertheless, these modifications are performed on affine subspaces, whereas modifying bent functions on suitable subsets may provide us with further classes of bent functions. In this article, we exactly specify new families of bent functions obtained by adding together indicators typical for the $\mathcal{C}$ and $\mathcal{D}$ class, thus essentially modifying bent functions in $\mathcal{M}$ on suitable subsets instead of subspaces. It is shown that the modification of certain bent functions in $\mathcal{M}$ gives rise to new bent functions which are provably outside the completed $\mathcal{M}$ class. Moreover, we consider the so-called 4-bent concatenation (using four different bent functions on the same variable space) of the (non)modified bent functions in $\mathcal{M}$ and show that we can generate new bent functions in this way which do not belong to the completed $\mathcal{M}$ class either. This result is obtained by specifying explicitly the duals of four constituent bent functions used in the concatenation. The question whether these bent functions are also excluded from the completed versions of $\mathcal{PS}$, $\mathcal{C}$ or $\mathcal{D}$ remains open and is considered difficult due to the lack of membership indicators for these classes

Categorified sl(N) invariants of colored rational tangles

We use categorical skew Howe duality to find recursion rules that compute categorified sl(N) invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress towards two conjectures about the colored HOMFLY homology of rational links.Comment: 45 pages, many figures, uses dcpic.sty, v2: minor changes and new example 5

The limits of Nečiporuk’s method and the power of programs over monoids taken from small varieties of finite monoids

Cotutelle avec l'École Normale Supérieure de Cachan, Université Paris-Saclay.Cette thèse porte sur des minorants pour des mesures de complexité liées à des sous-classes de la classe P de langages pouvant être décidés en temps polynomial par des machines de Turing. Nous considérons des modèles de calcul non uniformes tels que les programmes sur monoïdes et les programmes de branchement. Notre première contribution est un traitement abstrait de la méthode de Nečiporuk pour prouver des minorants, indépendamment de toute mesure de complexité spécifique. Cette méthode donne toujours les meilleurs minorants connus pour des mesures telles que la taille des programmes de branchements déterministes et non déterministes ou des formules avec des opérateurs booléens binaires arbitraires ; nous donnons une formulation abstraite de la méthode et utilisons ce cadre pour démontrer des limites au meilleur minorant obtenable en utilisant cette méthode pour plusieurs mesures de complexité. Par là, nous confirmons, dans ce cadre légèrement plus général, des résultats de limitation précédemment connus et exhibons de nouveaux résultats de limitation pour des mesures de complexité auxquelles la méthode de Nečiporuk n’avait jamais été appliquée. Notre seconde contribution est une meilleure compréhension de la puissance calculatoire des programmes sur monoïdes issus de petites variétés de monoïdes finis. Les programmes sur monoïdes furent introduits à la fin des années 1980 par Barrington et Thérien pour généraliser la reconnaissance par morphismes et ainsi obtenir une caractérisation en termes de semi-groupes finis de NC^1 et de ses sous-classes. Étant donné une variété V de monoïdes finis, on considère la classe P(V) de langages reconnus par une suite de programmes de longueur polynomiale sur un monoïde de V : lorsque l’on fait varier V parmi toutes les variétés de monoïdes finis, on obtient différentes sous-classes de NC^1, par exemple AC^0, ACC^0 et NC^1 quand V est respectivement la variété de tous les monoïdes apériodiques finis, résolubles finis et finis. Nous introduisons une nouvelle notion de docilité pour les variétés de monoïdes finis, renforçant une notion de Péladeau. L’intérêt principal de cette notion est que quand une variété V de monoïdes finis est docile, nous avons que P(V) contient seulement des langages réguliers qui sont quasi reconnus par morphisme par des monoïdes de V. De nombreuses questions ouvertes à propos de la structure interne de NC^1 seraient réglées en montrant qu’une variété de monoïdes finis appropriée est docile, et, dans cette thèse, nous débutons modestement une étude exhaustive de quelles variétés de monoïdes finis sont dociles. Plus précisément, nous portons notre attention sur deux petites variétés de monoïdes apériodiques finis bien connues : DA et J. D’une part, nous montrons que DA est docile en utilisant des arguments de théorie des semi-groupes finis. Cela nous permet de dériver une caractérisation algébrique exacte de la classe des langages réguliers dans P(DA). D’autre part, nous montrons que J n’est pas docile. Pour faire cela, nous présentons une astuce par laquelle des programmes sur monoïdes de J peuvent reconnaître beaucoup plus de langages réguliers que seulement ceux qui sont quasi reconnus par morphisme par des monoïdes de J. Cela nous amène à conjecturer une caractérisation algébrique exacte de la classe de langages réguliers dans P(J), et nous exposons quelques résultats partiels appuyant cette conjecture. Pour chacune des variétés DA et J, nous exhibons également une hiérarchie basée sur la longueur des programmes à l’intérieur de la classe des langages reconnus par programmes sur monoïdes de la variété, améliorant par là les résultats de Tesson et Thérien sur la propriété de longueur polynomiale pour les monoïdes de ces variétés.This thesis deals with lower bounds for complexity measures related to subclasses of the class P of languages that can be decided by Turing machines in polynomial time. We consider non-uniform computational models like programs over monoids and branching programs. Our first contribution is an abstract, measure-independent treatment of Nečiporuk’s method for proving lower bounds. This method still gives the best lower bounds known on measures such as the size of deterministic and non-deterministic branching programs or formulæ with arbitrary binary Boolean operators; we give an abstract formulation of the method and use this framework to prove limits on the best lower bounds obtainable using this method for several complexity measures. We thereby confirm previously known limitation results in this slightly more general framework and showcase new limitation results for complexity measures to which Nečiporuk’s method had never been applied. Our second contribution is a better understanding of the computational power of programs over monoids taken from small varieties of finite monoids. Programs over monoids were introduced in the late 1980s by Barrington and Thérien as a way to generalise recognition by morphisms so as to obtain a finite-semigroup-theoretic characterisation of NC^1 and its subclasses. Given a variety V of finite monoids, one considers the class P(V) of languages recognised by a sequence of polynomial-length programs over a monoid from V: as V ranges over all varieties of finite monoids, one obtains different subclasses of NC^1, for instance AC^0, ACC^0 and NC^1 when V respectively is the variety of all finite aperiodic, finite solvable and finite monoids. We introduce a new notion of tameness for varieties of finite monoids, strengthening a notion of Péladeau. The main interest of this notion is that when a variety V of finite monoids is tame, we have that P(V) does only contain regular languages that are quasi morphism-recognised by monoids from V. Many open questions about the internal structure of NC^1 would be settled by showing that some appropriate variety of finite monoids is tame, and, in this thesis, we modestly start an exhaustive study of which varieties of finite monoids are tame. More precisely, we focus on two well-known small varieties of finite aperiodic monoids: DA and J. On the one hand, we show that DA is tame using finite-semigroup- theoretic arguments. This allows us to derive an exact algebraic characterisation of the class of regular languages in P(DA). On the other hand, we show that J is not tame. To do this, we present a trick by which programs over monoids from J can recognise much more regular languages than only those that are quasi morphism-recognised by monoids from J. This brings us to conjecture an exact algebraic characterisation of the class of regular languages in P(J), and we lay out some partial results that support this conjecture. For each of the varieties DA and J, we also exhibit a program-length-based hierarchy within the class of languages recognised by programs over monoids from the variety, refining Tesson and Thérien’s results on the polynomial-length property for monoids from those varieties

Representation Theory and Analysis of Reductive Groups: Spherical Spaces and Hecke Algebras

The workshop gave an overview of current research in the representation theory and analysis of reductive Lie groups and its relation to spherical varieties and Hecke algebras. The participants and the speakers represented an international blend of senior researchers and young scientists at the start of their career. Some particular topics covered in the 30 talks related to structure theory of spherical varieties, p-adic symmetric spaces, symmetry breaking operators, automorphic forms, and local Langlands correspondence